Multiobjective optimization of an industrial wiped film poly(ethylene terephthalate) reactor: some further insights

Transcription

1 Computers and Chemical Engineering 25 (2001) Multiobjective optimization of an industrial wiped film poly(ethylene terephthalate) reactor: some further insights V. Bhaskar a, Santosh K. Gupta b,1, Ajay K. Ray a, * a Department of Chemical and Enironmental Engineering, National Uniersity of Singapore, 10 Kent Ridge Crescent, Singapore , Singapore b Department of Chemical Engineering, Uniersity of Wisconsin, Madison, WI 53706, USA Received 15 May 2000; received in revised form 5 December 2000; accepted 5 December 2000 Abstract Multiobjective optimization of an industrial third-stage, wiped-film poly(ethylene terephthalate) reactor is carried out, using a pre-validated model. The two objective functions minimized are the acid and vinyl end group concentrations in the product. These are two of the undesirable side products produced in the reactor. The optimization problem incorporates an end-point constraint to produce polymer having a desired value of the degree of polymerization (DP). In addition, the concentration of the di-ethylene glycol end group in the product is constrained to lie within a certain range of values. The possible decision variables for the problem are the reactor pressure, temperature, catalyst concentration, residence time of the reaction mass in the reactor and the speed of rotation of the agitator. The nondominated sorting genetic algorithm (NSGA) is used to solve this multiobjective optimization problem. It is found that this algorithm is unable to converge to the correct solution(s) when two or more decision variables are used, and we need to run the code several times over (with different values of the computational variable, S r,the seed for generating the random numbers) to obtain the solutions. In fact, this is an excellent test problem for future multiobjective optimization algorithms. It is found that when temperature is kept constant, Pareto optimal solutions are obtained, while, when the temperature is included as a decision variable, a global unique optimal point is obtained Elsevier Science Ltd. All rights reserved. Keywords: Multiobjective optimization; Polyethylene terephthalate; Genetic algorithm; Wiped-film reactor; Pareto sets; Fibre grade polyester Nomenclature a a o A A l b o DEG DP E a E DEG E g EG E v specific interfacial area per unit volume of the melt (m 1 ) coefficient in the polynomial relating 1 to polymer concentration acetaldehyde cross-sectional area of the melt in the reactor (m 2 ) coefficient in the polynomial relating 1 to polymer concentration di-ethylene glycol degree of polymerization, or the number average chain length of the polymer acid end groups DEG end groups (excluding those on pure DEG) hydroxyl end groups (excluding those on pure EG) ethylene glycol vinyl end groups * Corresponding author. Tel.: ; fax: address: cheakr@nus.edu.sg (A.K. Ray). 1 Present Address: Indian Institute of Technology, Kanpur , India /01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 392 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) I vector of objective functions, I i k 1 k 9 forward reaction rate constants for Eqs. (1) (9) in Table 1 k 1, k 5, k 7, reverse reaction rate constants for Eqs. (1), (5), (7) and (8) in Table 1 k 8 k 2o, k 9o frequency factors of reactions (Eqs. (2) and (9)) (Table 1), respectively k l overall liquid phase mass transfer coefficient (m min 1 ) K 1, K 5, K 7, equilibrium constants for Eqs. (1), (5), (7) and (8) in Table 1 K 8 l chrom chromosome length L length of the reactor (m) N speed of the agitator, (rpm) N* dimensionless value of N (N/N ref ) N g number of generations in NSGA N gen maximum number of generations in NSGA N p number of chromosomes in the population, in NSGA p c crossover probability in NSGA p m mutation probability in NSGA P pressure of the reactor (mmhg or kpa) P e penalty value in Eqs. (5a) and (5b) Q volumetric flow rate of liquid in the reactor, (m 3 min 1 ) S r seed for the random number generator in NSGA T temperature (K) u vector of decision variables, u i w 1, w 2, w 3 weighting factors used in the objective functions W water x vector of state variables, x i z dimensionless axial location in reactor (axial position/l) Z di-ester groups Greek alphabets exponent describing the effect of agitator speed on k l a sh exponent controlling the sharing effect in NSGA residence time (A l L/Q) (min) * dimensionless value of (/ ref ) maximum normalized distance in u space between any two points in NSGA 1 Flory parameter describing vapor liquid equilibrium Symbols [*] concentration in the melt, kmol m 3 (unless otherwise specified) Subscripts/ superscripts d f out 0 ref desired value feed value outlet value value at z=0 (inlet of reactor) reference conditions, as in Bhaskar et al. (2000a)

3 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Introduction Multiobjective optimization has created immense interest in engineering in the last two decades. Polymerization processes are quite complex in nature and offer themselves as excellent candidates for the application of multiobjective optimization. The quality of the polymer produced is normally described in terms of several experimentally measured indices, e.g. stiffness, tenacity, strength, etc. Fortunately, several of these experimental properties can be related to molecular parameters that can be predicted using mathematical models of the reactors. These include the average molecular weight, polydispersity index, degree of short-chain branching, concentration and distribution of functional groups, etc. The operating (and design) variables of a polymerization reactor system influence these molecular measures of the product in interesting and, often, conflicting ways. This is why multiobjective optimization is of considerable interest for polymerization systems. In this study, we focus our attention on the multiobjective optimization of polyethylene terephthalate (PET) reactors. PET, usually referred as polyesters, is one of the most widely used polymers in practice, and is extensively used in the manufacture of fibres, films, bottles, etc. A recent review on the applications of multiobjective optimization in chemical engineering (Bhaskar, Gupta & Ray, 2000b) suggests that several interesting studies have been reported on the multiobjective optimization of polymerization reactors. Tsoukas, Tirrell and Stephanopoulos (1982), Fan, Landis and Patel (1984), Farber (1986, 1989), Butala, Choi and Fan (1988), Choi and Butala (1991) carried out multiobjective optimization of copolymerization reactors. These early studies used the parametric method or the -constraint method (Chankong & Haimes, 1983) to obtain Pareto optimal sets of non-dominant solutions. A Pareto set is defined such that when we go from any one point to another, at least one objective function improves and at least one other worsens (Chankong & Haimes, 1983). Wajge and Gupta (1994) obtained optimal temperature histories corresponding to different points on the Pareto optimal sets for a non-vaporizing nylon-6 batch reactor, using similar techniques. Sareen and Gupta (1995) obtained optimal pressure histories and optimal alues of the jacket-fluid temperature at different points on the Pareto set for an industrial, semi batch nylon-6 reactor. In this reactor, monomer and water vaporize, and a control valve releases the vapors in a manner that a desired pressure history is maintained. More recently, our group (Chakravarthy, Saraf & Gupta, 1997; Mitra, Deb & Gupta, 1998; Garg & Gupta, 1999; Garg, Gupta & Saraf, 1999; Gupta & Gupta, 1999) has carried out multiobjective optimization studies on nylon-6 and polymethyl methacrylate (PMMA) reactors using the very robust genetic algorithm (GA). An adaptation of GA was used for multiobjective optimizations. This is the nondominated sorting genetic algorithm (NSGA), as developed by Srinivas and Deb (1995) for problems where the decision variables can take on optimal alues. This technique has been further adapted by Mitra et al. (1998) to apply for problems involving decision variables that are functions of time or space. Pareto sets of non-dominated solutions were obtained in all these studies. The Pareto sets usually obtained in multiobjective optimization help channelize the thinking of a decision-maker since they narrow down the choices from among which he or she can choose the preferred solution. This is usually done using intuition or experience, information that is often non-quantifiable (Chankong & Haimes, 1983). It may be emphasized that multiobjective optimization of real-life systems is quite complex and each new application may require the development of several adaptations of optimization algorithms to obtain meaningful solutions, irrespective of which mathematical procedure is used for the purpose. Indeed, a few such adaptations of GA have been developed in earlier studies (Goldberg, 1989; Deb, 1995). Recently, Bhaskar, Gupta and Ray (2000a) have carried out the multiobjective optimization of reactors producing fiber-grade PET. Indeed, any amount of lowering of costs indicated by such studies is of tremendous help in the polyester industry. Commercially, PET is manufactured in three stages, using continuous reactors. The first (esterification) stage is carried out at atmospheric pressure and at C. The raw materials commonly used are a molar excess of ethylene glycol (EG) and either purified terephthalic acid (PTA) or dimethyl terephthalate (DMT). Our study is based on the PTA route that is now quite popular. PTA and EG are usually processed in a series of CSTRs or a plug flow reactor with a recycle in the first stage. A polycondensation catalyst, antimony trioxide, is injected in small concentrations ( wt.%) into the oligomer stream leaving this reactor. The second (prepolymerization) stage is carried out either in one or two agitated vessels under reduced pressures, at about 2 4 kpa (15 30 mmhg) and C. The degree of polymerization (DP) attains a value of about in the second stage. The prepolymer so produced undergoes final polycondensation in a finishing (or third stage, wiped film) reactor in which the pressure is maintained quite low at kpa (1 2 mmhg), and temperatures are maintained at about C. Since the reaction mass is extremely viscous under these conditions, the finishing reactor has a special construction to enhance mass transfer and the removal of the by-product, ethylene glycol, so as to drive the reaction in the forward direction and to give a product having a high value of DP. The finisher is usually a jacketed

4 394 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) cylindrical vessel with a horizontal agitator, with large screens mounted on the latter. The reaction mass in the third-stage reactor is usually heated by condensing vapor in a jacket. Table 1 Kinetic scheme

5 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Fig. 1. Optimum values of [E a ] out vs. [E v ] out for several values of S r (Table 4) for the five-decision variable problem of Eqs. (4a), (4b), (4c), (4d) and (4e) (a). The points correspond to those given in Table 4. Corresponding values of the decision variables are also shown. Several studies on the modeling and simulation of the different stages of PET manufacture have been reported, as reviewed by Ravindranath and Mashelkar (1986a,b). In contrast, only a few studies (Kumar, Sharma & Gupta, 1984a,b; Kumar, Sukthankar, Vaz & Gupta, 1984; Immanuel & Gupta, 2000) have appeared in the open literature on the optimization of PET reactors. In our earlier study (Bhaskar, Gupta & Ray, 2001), we modified the model of Laubriet, LeCorre and Choi (1991) and simulated an industrial wiped film reactor. We tuned the model parameters using three sets of industrial data available with us. The tuned model, without any changes in the model parameters, predicted the fourth set of data extremely well. Thus, we believe that we have a model that represents the important physico-chemical phenomena in the reactor quite well, and that we could use it for optimizing its performance. Appendix A gives a summary of the model equations and parameters used (Bhaskar et al., 2000a; Bhaskar et al., 2001). There are two important objectives in this polymerization. These are the minimization of the acid and the vinyl end group concentrations, [E a ] and [E v ], respectively, in the product (represented by subscript, out ). The acid end group makes the polymer susceptible to hydrolysis (Besnoin & Choi, 1989) during the downstream operations and leads to breakage of the filaments during spinning, where the humidity is high. The vinyl end groups have been shown to be responsible for the coloration of PET (Besnoin & Choi, 1989; James & Packer, 1995) because of reactions not too well understood right now, and not included in our kinetic scheme shown in Table 1. Hence, the minimization of these two end groups improves the quality of the polymer product. The reduction of [E a ] simultaneously increases the rate of polymerization of the acid end group catalyzed polycondensation reaction, and helps maximize the throughput (this catalytic effect is also not directly incorporated in the model). The important end-point constraint is to produce polymer having a desired value of DP. An important side product is the diethylene glycol (DEG) end groups. Its presence affects the crystallinity and hence the melting point of the polymer unfavorably. However, these end groups im-

6 396 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) prove the dyeability of the fiber. Therefore, it is preferred to have a certain allowable range for the concentration of DEG end groups in the fiber-grade polyester produced. An inequality constraint is, therefore, imposed for the DEG end group concentration in the product. In addition, a further inequality constraint on the maximum allowable limit for the acid end group concentration is imposed to ensure that it is not only minimized, but also lies below an upper limit. In our earlier optimization study (Bhaskar et al., 2000a) of the wiped film PET reactor we used five decision variables, viz. the reactor pressure (P), temperature (isothermal, T), catalyst concentration ([Sb 2 O 3 ]), residence time () of the polymeric reaction mass inside the reactor, and the speed (N) of the wiped-film agitator. All of these variables can easily be changed in any industrial, wiped-film reactor for PET manufacture, including the one being studied here. We found that the multiobjective optimization problem described above had a unique optimum solution (no Pareto set of several equally good, non-dominating points was obtained). Also, we found that several minima in the decision-variable space were present, where the values of the two objective functions were almost identical (Fig. 1a) for all practical purposes. This result, though interesting, was somewhat unexpected and so we attempted to explore this problem further to develop more insight. This paper presents the results of this detailed study. 2. Modeling and simulation of the wiped-film reactor We give a short summary of the model used (Bhaskar et al., 2001) in this section. In the third-stage wiped-film PET reactor, the main reaction is polycondensation. This is accompanied by several other reactions, and the entire set of reactions (kinetic scheme) is given in Table 1 (Ravindranath & Mashelkar, 1984). The side reactions considered are the formation of hydroxyl end groups, acid end groups, vinyl end groups and DEG end groups. EG, water (W) and DEG are the volatile by-products of the process. We have used an adaptation of the two-phase reactor model of Laubriet et al. (1991), as described by Bhaskar et al. (2000a). The model (state variable) equations can be written in the form dx dz =f(x, u); x(z=0)=x 0 (1) where x is the vector of state variables, defined by x=[[e g ], [E a ], [Z], [E v ],[E DEG ], [EG], [W], [DEG]] T (2) and u is the vector of control or decision variables. In Eq. (1), z represents the (dimensionless) axial position in the wiped-film reactor. The model equations (a set of ordinary differential equations, ODEs and given in Appendix A) are solved using the IMSL subroutine, DIVPAG, which uses Gear s method (Gupta, 1995). The model provides values of DP and the concentrations of the hydroxyl end groups (E g ), acid end groups (E a ), di-ester end groups (Z), vinyl end groups (E v ), DEG end groups (E DEG ), EG, W and DEG, as a function of the axial position in the reactor. Model tuning and validation was performed using the industrial data given in Table 2. A tuning of nine parameters, k L a ref, a o, b o, K 1, K 5, K 8, k 2o, k 9o and, was carried out using three of the four sets (reference set 1, and sets 3 and 4) of industrial data given in Table 2. An IMSL subroutine DBCPOL (Nelder Mead method) was used. The following equation was used to relate k L a to the speed of the agitator, N, (with the volume of the melt remaining unchanged). k L a=k L a ref N (3) N ref In Eq. (3), subscript, ref, refers to the value of k L a at the reference conditions (set 1, Table 2). In the present case, the value of is expected to be higher than 0.5 (Higbie s penetration theory) as kinetic effects are coupled with mass transfer. The values of the tuned parameters are given in Table 2. It is observed from this Table 2 Industrial data and predicted values at the outlet end for the four data sets Set a Operating conditions Product property Industrial value Model predicted value b Reference (ref) a 1 DP out [E DEG ] out (kmol m 3 ) [E a ] out (kmol m 3 ) T ref +1K DP out P ref 0.5 mmhg DP out N ref +0.1 rpm DP out a T ref = K; P ref =2.0 mmhg; [Sb 2 O 3 ] ref =0.04 wt.%; *=1.0; N*=1.0; for the other three sets, only one of these values is different, as indicated. Sets 1, 3 and 4 used for tuning; set 2 used for validation. b Tuned values of parameters, k L a ref =2.6875; a o =1.0378; b o =2.1838; K 1 =6.1835; K 5 = ; K 8 =11.87; k 2o = ; k 9o = ; =

7 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Table 3 Computational variables used in this study/bounds and reference values of the possible decision variables Weighting factors (Eqs. (5a) and (5b)) GA parameters (ref) w 1 =w 3 =10 4 N gen =50 w 2 =10 8 N p =50 P e =10 4 p c =0.65 p m =0.004 = Variables/units sh =2.0 l chrom =32 bits Lower bound Upper bound Reference value P, kpa (mmhg) 0.05 (0.4) (2.0) (2.0) T (K) [Sb 2 O 3 ] (wt.%) * N* table that the model-predicted values of the product properties agree quite well with industrial values, not only for the three sets used for tuning (sets, 1, 3 and 4) but also for set 2, when these parameters are used. 3. Multiobjective optimization Five decision variables were used for multiobjective optimization in our previous study (Bhaskar et al., 2000a). These are P, T, [Sb 2 O 3 ],, and N. The multiobjective function optimization problem described earlier is described mathematically by MinI(P, T, [Sb 2 O 3 ], *, N*)[I 1, I 2 ] T =[[E a ] out,[e v ] out ] T subject to (4a) DP out =DP d (4b) [E a ] out kmol m 3 (4c) [E DEG ] out 0.17 kmol m 3 (4d) dx dz =f(x, u); x(z=0)=x 0 (4e) In Eqs. (4a), (4b), (4c), (4d) and (4e), subscripts out and d refer to the values at the outlet of the reactor and the desired values of the product property, respectively. The variables, * andn*, represent dimensionless values, / ref and N/N ref, where ref and N ref (reference values) are the values being used currently in the industrial reactor being studied. These values are confidential and are not being provided due to proprietary reasons. Meaningful bounds have been chosen on the five decision variables, u, based on industrial practice. These are given in Table 3. A few important points need to be mentioned here. We have used the catalyst concentration as a decision variable for the optimization of stage 3, even though the catalyst is added (in the form of the glycolate) at the inlet of stage 2. This has been done because the catalyst concentration used affects the operation of stage 3 far more significantly than stage 2, and remains unconsumed. The feed to the wiped-film reactor is assumed to be constant while carrying out the optimization, even though we have taken * as a decision variable. A different value of * would require a higher flow rate in the wiped-film reactor, as well as in the previous reactors (we assume that a change in * is achieved by a change in the flow rate in the wiped-film reactor, rather than the liquid hold up). This is justified since stage 2 is equilibrium-controlled (Ravindranath & Mashelkar, 1982), and its output concentrations depend only on its temperature and pressure, and are independent of the flow rate in it. Both the equality and the inequality constraints (Eqs. (4b), (4c) and (4d)) are incorporated in the objective functions as penalty functions and they can be represented mathematically as Min I*=w 1 1 [E a ] out +w 2 DP 1 out 2 +P e (5a) DP d Min I*=w 2 3 [E v ] out +w 2 DP out 2 1 +P e (5b) DP d An additional large penalty value, P e (=10 4 ), is added to the objective functions, I 1 * and I 2 *, if either of the two inequality constraints in Eqs. (4c) and (4d) are violated. If these constraints are satisfied, then the value of P e in Eqs. (5a) and (5b) is taken as zero. This ensures that bad chromosomes in the genetic algorithm used for optimization do not get reproduced in the successive generations even if several chromosomes violating these constraints exist in the initial population (i.e. the chro-

8 398 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) mosomes violating Eqs. (4c) and (4d) are killed almost instantaneously using P e ). Minimization of I 1 * and I 2 * leads to a decrease in the acid and vinyl end group concentrations, while simultaneously giving preference to solutions satisfying the several requirements discussed above. The use of a penalty to kill chromosomes when important physical constraints are violated is one of the adaptations made in the basic algorithm (NSGA) in this study. 4. Results and discussion A computer program was written in FORTRAN 90 and debugged using several test problems (Deb, 1995). The solution of the multiobjective optimization problem described in Eqs. (4a), (4b), (4c), (4d) and (4e) was then obtained using the computational parameters and bounds of the decision variables given in Table 3. The CPU time taken for one optimization run in SGI Origin 2000 super computer varied from to s depending on the number of decision variables used. The value of DP d was taken as 82.0, the same as the value for the reference case (set 1 in Table 2) for the industrial reactor simulated earlier. This enables one to explore if changes in the operating variables could improve the performance of the existing industrial reactor (which was simulated by us, Bhaskar et al., 2001). A unique optimal point (solution) was obtained, using S r = (where S r is the seed used for generating random numbers in the code) as given below (Bhaskar et al., 2000a). [E a ] out = kmol m 3 (6) [E v ] out = kmol m 3 The corresponding values of the decision variables are P=0.18 kpa (1.39 mmhg); T= K; [Sb 2 O 3 ]= wt.%; (7) *=0.967; N*=1.0 This point is shown as point 4 in Fig. 1. We then ran our computer program for several values Table 4 Values of S r used for the five-decision variable problem c a a (ref) a Multiplicity observed in these cases. Correspond to best optimal solutions. S r of S r. Unique points were always obtained, irrespective of the value of S r. The results are shown in Fig. 1 for six values of S r given in Table 4. It is observed from Fig. 1a that NSGA converges to different (though unique) local optimal points, and is unable to converge, in general, to the global optimal solution (point 4 (obtained using S r =0.8887), and given in Eqs. (6) and (7)). The failure of NSGA for this multiobjective optimization problem contrasts with the almost overwhelming success of this technique for several other complex Chemical Engineering problems studied by our group recently (Bhaskar et al., 2000b; Chan, Aatmeeyata, Gupta & Ray, 2000). Indeed, Goldberg (1989) has indicated that even though GAs are quite efficient in reaching the global-optimum region, they are not guaranteed to reach the precise location of the global-optimum point(s), particularly for complex problems. The PET wiped film reactor optimization problem, thus, could be an excellent example for testing newer multiobjective optimization algorithms. It may be added that NSGA/GA has certain advantages over other popular gradient-based techniques like NLP/ SQP. Lexicographic approach works on the principle of ranking the objective function based on the importance a priori. The disadvantage with lexicographic approach is that the solution is very sensitive to the ranking of the objectives by the decision-maker (DM) and, therefore, one should be careful in applying this technique to objectives having almost equal importance. Different results may be obtained for such cases depending upon the ranking of such objectives by the DM (Chankong & Haimes, 1983). Ranking of objective functions a priori is also a difficult task. The evolutionary algorithms are far superior compared with gradient search techniques in handling discrete search space problems. NSGA has been found to be a highly robust and successful technique for obtaining the optimal solutions for several complex multi-objective optimization problems in chemical engineering. Interestingly, it fails for the PET reactor problem unless adaptation as mentioned in this manuscript is used. An interesting point suggested by Fig. 1 is that even for near-identical values of the two objective functions, [E a ] out and [E v ] out, as for example, points 2 and 4 (corresponding to values of S r of and ), the optimal values of the decision variables differ significantly. This reflects the nature of the terrain and indicates the presence of multiple (local) minima having almost equal depths. The presence of such multiple optimal points and the failure of NSGA to give the global (or all the) optimal solution(s) in a single application, are to be noted since this could happen in other optimization problems as well. We also solved modified versions of the multiobjective optimization problem described in Eqs. (4a), (4b), (4c), (4d) and (4e), which had fewer decision variables. This was done to develop further insight into the multiobjective optimization problem. In the first such problem, we

9 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) N*=C 4 (8) The bounds on P are taken to be the same as in Table 3. Unique optimal solutions were found for this problem, too, for all the choices of the constants, C 1 C 4. Interestingly, use of different values of S r for any specified values of C 1 C 4, did not give different optimal solutions in this case, in contrast to what we experienced with Eqs. (4a), (4b), (4c), (4d) and (4e). Fig. 2 (filled circles) shows the unique optimal solutions of Eq. (8) for [Sb 2 O 3 ]=0.04 wt.%, *=1 and N*=1.0, i.e. for several different (constant) values of T in the range KT K. Each constant value of T gives a single unique optimal Fig. 2. Optimum solutions for a single-decision variable (P) optimization problem (Eq. (8)) at different values of T (filled circles). Unfilled circles represent the optimal solutions for the two-decision variable (P, T) problem (Eq. (9)) for different S r.[sb 2 O 3 ]=0.04 wt.%; *= 1.0; N*=1.0 for both cases. used only one decision variable, the pressure, P, and kept the other decision variables at constant values (mostly, at their reference values given in Table 3). The modified problem is described mathematically by Min I(P)[I 1, I 2 ] T =[[E a ] out,[e v ] out ] T s.t. Eqs. (4b), (4c), (4d) and (4e) and T=C 1 [Sb 2 O 3 ]=C 2 *=C 3 Fig. 3. Optimum solutions for a single-decision variable (P) optimization problem (Eq. (8)) at (filled circles) and K (unfilled triangles), for different constant values of [Sb 2 O 3 ]. Unfilled circles represent the optimal solutions for the two-decision variable (P, [Sb 2 O 3 ]) optimization problem for different S r. *=1.0; N*=1.0 for all cases.

10 400 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) s.t. Eqs. (4b), (4c), (4d) and (4e) and [Sb 2 O 3 ]=C 2 =0.04% *=C 3 =1.0 N*=C 4 =1.0 (9) with bounds on P and T as given in Table 3. Fig. 2 (open circles) gives the optimal solutions for this problem for different values of S r. We obtain unique solutions for each value of S r. This means that as soon as we move from a one-decision variable problem to one Fig. 4. Optimum solutions for the single-decision variable (P) optimization problem (Eq. (8)) (filled circles) for different constant values of *. Unfilled circles represent the optimal solutions for the two-decision variable (P, *) optimization problem for different S r. T= K; [Sb 2 O 3 ]=0.04 wt.%; N*=1.0 for all cases. solution in the [E a ] out versus [E v ] out, plane. This diagram shows that temperature affects both [E a ] out and [E v ] out in a similar manner. This is expected since both [E a ] out and [E v ] out are products of degradation reactions. Clearly, point A in Fig. 2a is the best point among all those obtained, dominating over all the other points shown. We should, therefore, expect to obtain point A as the unique global optimum solution if we use NSGA to solve the following multiobjective optimization problem with two decision variables, P and T. Min I(P, T)[I 1, I 2 ] T =[[E a ] out,[e v ] out ] T Fig. 5. Optimum solutions for the single-decision variable (P) optimization problem (Eq. (8)) (filled circles) for different constant values of N*. Unfilled circles represent the optimal solutions for the two-decision variable (P, N*) optimization problem for different S r. T= K; [Sb 2 O 3 ]=0.04 wt.%; *=1.0 for all cases.

11 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Fig. 6. Optimum solutions of several problems not involving T as a decision variable. The decision variables used are: filled circles, P, N*; unfilled circles, P, *; unfilled triangles, P, *, N*; unfilled squares, P, [Sb 2 O 3 ], *, N*; all for T= K. Filled squares are for T= K with decision variables as P, [Sb 2 O 3 ], *, N*. Reference values (Table 3) used for the operating variables held constant. involving two decision variables in the PET wiped film reactor optimization problem, NSGA fails to give the global optimal point in a single application. However, multiple applications of this technique with several different values of S r could be used to obtain the complete and correct solution. We now study the one decision variable optimization problem in Eq. (8) with T= K, *=1.0, N*=1.0 and with different (but constant) values of [Sb 2 O 3 ]. Fig. 3 (filled circles) shows the results. The unfilled triangles in Fig. 3 show similar results, but for T=C 1 = K. In contrast to what was observed in Fig. 2, Pareto sets are obtained at both the temperatures studied (as well as at other intermediate values of temperature, not shown in Fig. 3). The Pareto for T= K is superior to that for T= K. We used NSGA to solve the two objective function problem with T= K, and with P and [Sb 2 O 3 ] as the decision variables. Again, a unique optimal solution was obtained for any value of S r, instead of the expected Pareto set, indicating the failure of NSGA. Optimal solutions were generated for the two-decision variable problem using several values of S r,and these are shown (unfilled circles) in Fig. 3. Interestingly, these solutions superpose quite well with those obtained at T= K using one decision variable, and suggest a means of obtaining meaningful solutions with NSGA (by running NSGA for several values of S r ). Figs. 4 and 5 show the optimal solutions (filled circles) for the single-decision variable (P) problem (Eq. (8)), for different values of * (Fig. 4) and N* (Fig. 5). Each run leads to a unique solution, but by varying the appropriate decision variable, we obtain Pareto sets for these two

12 402 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) cases too, as in Fig. 3. The dominating Pareto, once again, was found to correspond to T= K (Paretos were obtained for higher values of T too, but are not shown here). For this value of T, NSGA is used to obtain the optimal solutions for the two-decision variable multiobjective optimization problem (P and * for Fig. 4; P and N* for Fig. 5). Once again, NSGA gives only unique points as the optimal solution, and is unable to catch the Pareto sets. Only when we use several values of S r, are we able to obtain the Paretos indicated by the simpler and equivalent one decision variable problem. It appears (from Figs. 3 5) that running NSGA several times with different S r helps in obtaining solutions of the PET reactor optimization problem, at least for the three sets of the two-decision variable problems. Based on our observations in Figs. 2 5 we infer that the multiobjective optimization problem with a single decision variable (Eq. (8)) offers the greatest amount of insight in understanding the effects of the decision variables on the objective functions. Also, it is safest to start from optimal solutions of these simple problems, and then attempt to generate optimal solutions to problems involving more than one decision variable. NSGA can still be used to generate solutions for the present problem, but only through the use of several applications with different values of S r (each of which gives only a single, unique optimal point rather than the entire Pareto set, even if the latter is the correct solution). It is interesting to note that we obtain Pareto solutions for all the two decision variable multiobjective optimization problems studied until now, except for the first case Fig. 7. Optimum solutions of several multiobjective optimization problems. The decision variables used are: filled circles, P, T; unfilled circles, P, N*; unfilled triangles, P, T, N*; unfilled squares, P, T, *, N*. [Sb 2 O 3 ]=0.04 wt.% in all cases. Reference values (Table 3) used for the operating variables held constant.

13 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Fig. 8. Optimum solutions for the two-decision variable (P, N*) optimization problem at three different values of T. [Sb 2 O 3 ]=0.04 wt.%; *=1.0. when P and T are the decision variables, where a unique global optimum solution is indicated. Yet another interesting point to note is that Figs. 2 5 do not show much scatter, something that confounded us in our earlier optimization study of this reactor (Bhaskar et al., 2000a). We now present solutions of a few multiobjective optimization problems with three and four decision variables. In these cases, too, NSGA fails to catch the Pareto solutions or the dominating solution, and gives only single optimal points, depending on the value of S r used. Fortunately, in these cases too, multiple applications of NSGA with different values of S r gives the complete and correct picture. Fig. 6 shows the solutions obtained using two, three and four decision variables. None of these sets of decision variables involved T, which was kept constant at K. It is observed that as soon as the number of decision variables increases beyond two, scatter is manifested in the plots of the decision variables, even though there is a reasonable amount of overlap (and only a small amount of scatter) in the Pareto solutions obtained in all the cases. The scatter is particularly large in the plots of the optimal values of P and * when three or four decision variables are used for optimization. This makes such plots very inconvenient to use in industry. The scatter is associated with the fact that the decision variables compensate for each other and give similar values of [E a ] out and [E v ] out. Fig. 6 shows that we also obtain Pareto sets (using several runs of NSGA with different values of S r ) at a constant temperature of K. Based on the Pareto sets obtained with four decision variables, P, [Sb 2 O 3 ], *, and N*, at the two temperatures ( and K), we expect that if we use all five decision variables, we would obtain the same optimal solution as the Pareto at K and with four decision variables. However, this does not happen and the moment we add on temperature as the fifth decision variable, the Pareto vanishes (see Fig. 1) and we have a global optimum point. Fig. 7 shows another set of results with several sets of decision variables. This time, the temperature is included as one of the decision ariables, in (almost) all cases. In this case, Paretos are definitely not obtained (we believe that this is the effect of including T in the set of decision variables), and a single dominating optimal point exists. However, NSGA is again unable to converge to this correct solution in any of the cases and gives single but incorrect optimal solutions. Several runs with different values of S r were made again to seek out the correct unique solution. Fig. 7 shows that a considerable amount of scatter exists in the optimal solutions when four decision variables are used. It is clear that the effect of temperature is most dominant whenever it is included as a decision variable, and unique optimal points are obtained in all such cases. Lower temperatures lead to lower values of both [E a ] out and [E v ] out, and a unique optimal solution is expected physically. However, we were unable to rationalize all these observations earlier (Bhaskar et al., 2000a) because of lack of sufficient insight developed here. It is to be emphasized that NSGA was unable to obtain either Pareto optimal solutions or the unique global optimal point, in a single application in the present study. This is in sharp contrast to our earlier studies on multiobjective optimization of steam reformers (Rajesh, Gupta, Rangaiah & Ray, 2000), cyclone separators (Ravi, Gupta & Ray, 2000)and PMMA reactors (Zhou, Gupta & Ray, 2000), systems that are equally complex. We did solve several other (simpler) multiobjective optimization test problems (Deb, 1995) with the same code, and were able to obtain the correct Pareto solutions in a single run. This confirms that our code is free of errors. It appears that the present multiobjective optimization problem is an excellent test problem for checking the efficiency of future optimization algorithms and adaptations.

14 404 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) It must be added that we found two solutions with almost identical values of [E a ] out and [E v ] out (points 2 and 4 in Fig. 1) for the optimization problem with five decision variables, but having very different values of the decision variables. We were unable to obtain similar multiple solutions with almost equal values of the objective functions, in problems involving three or four decision variables, even though we made several trial runs. Since we obtained Pareto sets of non-dominated solutions for several cases, we decided to develop a few correlations that could be used for industrial application (for the reactor being studied). We selected three problems, each involving two decision variables, P and N*. This was done because multiobjective optimization Fig. 10. Optimum solutions for the two-decision variable (P, N*) optimization problem at three different values of *. T= K; [Sb 2 O 3 ]=0.04 wt.%. Fig. 9. Optimum solutions for the two-decision variable (P, N*) optimization problem at three different values of [Sb 2 O 3 ]. T= K; *=1.0. problems having more than two decision variables showed considerable scatter, and fitting correlations would be meaningless. The reason why we selected P and N* was because these were the most commonly varied decision/operating variables used in industrial wiped-film PET reactors. The results are shown in Figs The Pareto sets are observed to be well described by straight lines in all these cases. Table 5 gives the correlations. Once the decision-maker decides upon a preferred solution (a preferred value of, say, the acid end group concentration) and selects say, a reactor temperature, the correlations given in Table 5 could be used to obtain the optimal values of pressure, N* and the vinyl end group concentration.

15 V. Bhaskar et al. / Computers and Chemical Engineering 000 (2001) Conclusions An improved and validated model for the wiped-film finishing reactor for PET manufacture is used to solve an optimization problem involving two objective functions and some end-point constraints. The concentrations of the side products, [E a ] out and [E v ] out, are minimized, while ensuring that the product has certain desired characteristics (DP out, [E a ] out and [E DEG ] out,). The highly successful technique (based on our earlier experiences), NSGA, fails to give correct solutions in a single application, particularly when several decision variables are used. One needs to study several simpler optimization problems with fewer decision variables, and also apply NSGA several times over (using different values of S r ) in order to be able to obtain the correct solutions. It is found that for the PET wiped film reactor, Pareto solutions of non-dominating solutions are obtained whenever temperature is kept constant and is not used as a decision variable. In contrast, a globally unique optimal point is obtained when the temperature is taken as one of the decision variables, because its effect on the two objective functions dominates. Appendix A. Complete set of model equations used in this study Balance equations for the liquid phase (Laubriet et al., 1991) 1 d[e g ] dz =[ 2R 1 R 2 R 3 R 4 R 5 2R 6 +R 7 R 8 ] 1 d[e a ] dz =[R 2+R 4 +R 6 R 7 R 8 +R 9 ] 1 d[z] dz =[R 1+R 3 +R 5 +R 8 R 9 ] 1 d[e v ] dz =[ R 3+R 9 ] 1 1 d[e DEG ] =[ R dz 5 +R 6 ] d[eg] =R 1 R 4 R 7 k l a([eg] [EG*])] dz 1 d[w] dz =R 7+R 8 k l a([w] [W *])] 1 d[deg] =R 4 +R 5 k l a([deg] [DEG*])] dz where R 1 =k 1 [E g ] 2 4k 1 [Z][EG] R 2 =k 2 [E g ] R 3 =k 3 [E v ][E g ] R 4 =2k 4 [E g ][EG] R 5 =k 5 [E g ][E DEG ] 4k 5 [Z][DEG] R 6 =k 6 [E g ] 2 R 7 =2k 7 [E a ][EG] k 7 [E g ][W] R 8 =k 8 [E g ][E a ] 2k 8 [Z][W] R 9 =k 9 [Z] Vapor liquid equilibrium correlations. C C*= poly j x *; j C*=[EG*], j [W*], [DEG*] * 1 x j j Table 5 Correlations for predicting optimal pressure, speed of the agitator and vinyl-end group concentrations for DP d =82.0 a Decision Variables Correlations T (K) [Sb 2 O 3 ] (wt.%) * P opt =1.393 T 3.666[E a ] out N* opt = [E a ] out T [E v ] out,opt = T [E a ] out P opt = [Sb 2 O 3 ] [E a ] out N* opt =37.69[Sb 2 O 3 ] [E a ] out [E v ] out,opt = [Sb 2 O 3 ] [E a ] out P opt = * [E a ] out N* opt =0.8404* [E a ] out [E v ] out,opt = 0.78* [E a ] out a P in mmhg; [E a ] out,[e v ] out in kmol m 3.

16 406 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) C poly = [E g]+[e a ]+[E v ]+[E DEG ] 2 x j *= Py j P jo j ; ln P o EG j=eg, W, DEG ln T = T ln P Wo = T 33.3 o ln P DEG = T y j = j k l a(c j C j *)dz 1 k l a(c j C*)dz+ j (k 2 [E g ]+k 3 [E v ][E g ])dz j=eg, W, DEG j = 1 m j exp 1 1 m j + 1 ; m j = V p j ; j=eg, W, DEG M j M U j C j v = ; j=eg, W, DEG j j V p = 1 U v C poly 1 =a o +b o U 1 v U v o k l a=k l a ref N N ref 0 j=eg, W, DEG DP= ([E g]+[e a ]+[E v ]+[E DEG ]+2[Z]) feed [E g ]+[E a ]+[E v ]+[E DEG ] Kinetic parameters (Laubriet et al., 1991; Martin & Choi, 1991) ([Sb 2 O 3 ]=0.04 wt.%) +. Reaction Type of Rate constant k i = Equinumber, i reaction k io exp( E i /RT) librium constant k io (m 3 E i (J mol 1 mol 1 ) min 1 ) 1 Re Irre * 3 Irre Irre Re Irre Re Re Irre * +, Values of K 1, K 5, K 8, k 2o and k 9o are those obtained in the present study, and differ from those in Laubriet et al. (1991); Martin and Choi (1991). *, min 1. References Besnoin, J. M., & Choi, K. Y. (1989). Identification and characterization of reaction byproducts in the polymerization of polyethylene terephthalate. Journal of Macromolecular Science Reiews Macromolecular Chemistry and Physics, C29, 55. Bhaskar, V., Gupta, S. K., & Ray, A. K. (2001). Modeling of an industrial wiped film poly(ethylene terephthalate) reactor. Polymeric Reaction Engineering, 9, in press. Bhaskar, V., Gupta, S. K., & Ray, A. K. (2000a). Multiobjective optimization of an industrial wiped film PET reactor. American Institute of Chemical Engineering Journal, 46(5), Bhaskar, V., Gupta, S. K., & Ray, A. K. (2000b). Applications of multiobjective optimization in chemical engineering. Reiews in Chemical Engineering, 16(1), 1. Butala, D., Choi, K. Y., & Fan, M. K. H. (1988). Multiobjective dynamic optimization of a semibatch free-radical copolymerization process with interactive CAD tools. Computers and Chemical Engineering, 12, Chakravarthy, S. S. S., Saraf, D. N., & Gupta, S. K. (1997). Use of genetic algorithms in the optimization of free radical polymerizations exhibiting the Trommsdorff effect. Journal of Applied Polymer Science, 63, 529. Chan, C. Y., Aatmeeyata, Gupta, S. K., & Ray, A. K. (2000). Multiobjective optimization of membrane separation modules. Journal of Membrane Science, 176, 177. Chankong, V., & Haimes, Y. Y. (1983). Multiobjectie decision making theory and methodology. New York: Elsevier. Choi, K. Y., & Butala, D. N. (1991). An experimental-study of multiobjective dynamic optimization of a semibatch copolymerization process. Polymer Engineering and Science, 31, 353. Deb, K. (1995). Optimization for engineering design: algorithms and examples. New Delhi: Prentice Hall. Fan, L. T., Landis, C. S., & Patel, S. A. (1984). In L. K. Doraiswamy, & R. A. Mashelkar, Frontiers in chemical reaction engineering (p. 609). New Delhi: Wiley. Farber, J. N. (1986). Steady state multiobjective optimization of continuous copolymerization reactors. Polymer Engineering and Science, 26, 499. Farber, J. N. (1989). In N. P. Cheremisinoff, Handbook of polymer science and technology, vol. 1 (p. 429). New York: Marcel Dekker.

17 V. Bhaskar et al. / Computers and Chemical Engineering 25 (2001) Garg, S., & Gupta, S. K. (1999). Multiobjective optimization of a free radical bulk polymerization reactor using genetic algorithm. Macromolecular Theory of Simulation, 8, 46. Garg, S., Gupta, S. K., & Saraf, D. N. (1999). On-line optimization of free radical bulk polymerization reactors in the presence of equipment failure. Journal of Applied Polymer Science, 71, Goldberg, D. E. (1989). Genetic algorithms in search, optimization and machine learning. Reading, MA: Addision-Wesley. Gupta, S. K. (1995). Numerical methods for engineers (p. 1995). New Delhi: New Age International. Gupta, R. R., & Gupta, S. K. (1999). Multiobjective optimization of an industrial nylon-6 semibatch reactor system using genetic algorithm. Journal of Applied Polymer Science, 73, 729. Immanuel, C. D., & Gupta, S. K. (2000). Optimization of the first stage continuous reactor-sequence in polyester manufacture from purified terephthalic acid. Journal of Polymer Engineering, 20, 51. James, D. E., & Packer, L. G. (1995). Effect of reaction time on poly(ethylene terephthalate) properties. Industrial Engineering and Chemical Research, 34, Kumar, A., Sharma, S. N., & Gupta, S. K. (1984a). Optimization of polycondensation step of PET formation with continuous removal of condensation products. Polymer Engineering Science, 24, Kumar, A., Sharma, S. N., & Gupta, S. K. (1984b). Optimization of polycondensation stage of PET reactor. Journal of Applied Polymer Science, 29, Kumar, A., Sukthankar, V. K., Vaz, C. P., & Gupta, S. K. (1984). Optimization of the transesterification of PET reactor. Polymer Engineering and Science, 24, 185. Laubriet, C., LeCorre, B., & Choi, K. Y. (1991). Two-phase model for continuous final stage melt polycondensation of poly(ethylene terephthalate). 1. Steady-state analysis. Industrial Engineering and Chemical Research, 30, 2. Martin, H.S., & Choi, K.Y., (1991). Two-phase model for continuous final stage melt Polycondensation of Poly(ethylene terephthalate). 2 Dynamic analysis. Industrial Engineering Chemistry Research, 30, Mitra, K., Deb, K., & Gupta, S. K. (1998). Multiobjective dynamic optimization of an industrial nylon 6 semibatch reactor using genetic algorithm. Journal of Applied Polymer Science, 69, 69. Rajesh, J. K., Gupta, S. K., Rangaiah, G. P., & Ray, A. K. (2000). Multiobjective optimization of steam reformer performance using genetic algorithm. Industrial Engineering and Chemical Research, 39, 706. Ravi, G., Gupta, S. K., & Ray, M. B. (2000). Multiobjective optimization of cyclone separators. Industrial Engineering and Chemical Research, 39, Ravindranath, K., & Mashelkar, R. A. (1982). Modeling of polyethylene terephthalate reactors: 5. A continuous prepolymerization process. Polymer Engineering and Science, 22, 619. Ravindranath, K., & Mashelkar, R. A. (1984). Finishing stages of PET synthesis: a comprehensive model. American Institute of Chemical Engineering Journal, 30, 415. Ravindranath, K., & Mashelkar, R. A. (1986a). Polyethylene terephthalate I. Chemistry, thermodynamics and transport properties. Chemical Engineering and Science, 41, Ravindranath, K., & Mashelkar, R. A. (1986b). Polyethylene terephthalate II. Engineering analysis. Chemical Engineering and Science, 41, Sareen, R., & Gupta, S. K. (1995). Multiobjective optimization of an industrial semibatch nylon 6 reactor. Journal of Applied Polymer Science, 58, Srinivas, N., & Deb, K. (1995). Multiobjective function optimization using nondominated sorting genetic algorithms. Eolutionary Computation, 2, Tsoukas, A., Tirrell, M. V., & Stephanopoulos, G. (1982). Multiobjective dynamic optimization of semibatch copolymerization reactors. Chemical Engineering and Science, 37, Wajge, R. M., & Gupta, S. K. (1994). Multiobjective dynamic optimization of a non-vaporizing nylon-6 batch reactor. Polymer Engineering and Science, 34, Zhou, F. B., Gupta, S. K., & Ray, A. K. (2000). Multiobjective optimization of the continuous casting process for poly(methyl methacrylate) using adapted genetic algorithm. Journal of Applied Polymer Science, 78,